MATH 1104 · Probability and Statistics: Discrete Random Variables

Led by F.N. David Simulacrum

Probability distributions, permutations and combinations, and the named distributions — Bernoulli, Binomial, Poisson, Geometric — through the history of how they were discovered.

1. Module 1

   Discrete Probability Distributions

   Led by F.N. David Simulacrum

   The question

   A random variable X takes values 1, 2, 3, 4. What makes a function a valid probability distribution — and why is the expected value called "expected" when it is not necessarily the most likely outcome?

   Outcome

   The student can construct a probability distribution and compute expected value, variance, and cumulative probabilities.

   Sub-units

   1. ○ 1.1 Build a Probability Distribution Open this unit → ✓ ↻
   Begin the course → Print certificate of completion
2. Module 2

   Transforming and Combining Random Variables

   Led by F.N. David Simulacrum

   The question

   The final score is 0.4×Test1 + 0.6×Test2. Test scores are independent. What is the expected final score and its standard deviation — and why does Var(X + Y) = Var(X) + Var(Y) require independence?

   Outcome

   The student can apply linear transformation rules for means and variances of combined random variables.

   Sub-units

   1. ○ 2.1 Transformation and Combination Open this unit → ✓ ↻
   Open this module → Print certificate of completion
3. Module 3

   Permutations, Combinations, and Counting

   Led by F.N. David Simulacrum

   The question

   Committee of 4 from 9 people: combinations. Top 3 in a race of 8: permutations. 5-card hand from 52: combinations. For each — why that choice, and what is the numerical answer?

   Outcome

   The student can compute permutations and combinations and select the appropriate method.

   Sub-units

   1. ○ 3.1 Counting Problems Open this unit → ✓ ↻
   Open this module → Print certificate of completion
4. Module 4

   Named Distributions: Bernoulli, Binomial, Poisson, Geometric

   Led by F.N. David Simulacrum

   The question

   12 coin flips, P(at least 9 heads). 4 calls per minute, P(exactly 6). Hit probability 0.3, P(first hit on 5th shot). Three distributions, three conditions to verify. Which applies — and why?

   Outcome

   The student can identify and apply Bernoulli, Binomial, Poisson, and Geometric distributions.

   Sub-units

   1. ○ 4.1 Named Distribution Problems Open this unit → ✓ ↻
   Open this module → Print certificate of completion
5. Module 5

   Putting the Distributions Together

   Led by F.N. David Simulacrum

   The question

   Probability theory is the child of gambling — but the Poisson distribution governs radioactive decay, web server traffic, and football scores. Choose one named distribution: trace its historical origin, then show its modern application in a completely unrelated field.

   Outcome

   The student can select the appropriate distribution and explain the universality of its mathematical structure.

   Sub-units

   1. ○ 5.1 Final Essay: The History of Chance Open this unit → ✓ ↻
   Open this module → Print certificate of completion